We construct a degree theory for Vanishing Mean Oscillation functions in metric spaces, following some ideas of Brezis & Nirenberg. The underlying sets of our metric spaces are bounded open subsets of ${ℝ}^N$ and their boundaries. Then, we apply our results in order to analyze the surjectivity properties of the $L$-harmonic extensions of VMO vector-valued functions. The operators $L$ we are dealing with are second order linear differential operators sum of squares of vector fields satisfying the hypoellipticity...

On obtient ici le développement asymptotique, en temps petit et sur la diagonale, du noyau de la chaleur associé à un opérateur dégénéré du second ordre satisfaisant à la condition forte d’hypoellipticité de Hörmander.

Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$. In this paper we study the Dirichlet problem for $ℒ$ with $L^p$-boundary data, on domains $\Omega$ which are contractible with respect to the natural dilations of $G$. One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$. A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as $t^{-p/2}$) and the Schrödinger equation (decay as $t^{(1-p)/2}$), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.